# Combinatorics of canonical bases revisited: String data in type $A$

## TL;DR

This paper provides explicit formulas and descriptions for the crystal structure and involutions on string data in type A, offering new insights into the combinatorics of canonical bases and inequalities for string polytopes.

## Contribution

It introduces a formula for the crystal structure on integer points of string polytopes and describes the Kashiwara *-involution explicitly for type A.

## Key findings

- Explicit crystal structure formula for string polytopes.
- Defining inequalities for Nakashima-Zelevinsky string polytopes.
- Explicit description of Kashiwara *-involution in type A.

## Abstract

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.